Rock Scully, in his book "Living With The Dead", gives an account of how the names were invented:

"Finally, Jerry and Healy fuse Anthem Of The Sun into one long continuous work.
It's a miracle of ingenuity, but I suddenly realise that unless we break it into tracks, we aren't going to get paid more than two cents for publishing rights for the whole record.
And so, after they have achieved the near impossible by putting together these thousands of scraps of tape into a coherent seamless whole, I have to tell Healy and the rest of the group that we have to take the thing apart - band it.
For the purposes of publishing (and airplay) there's got to be moments of silence between "Cryptical Envelopment" and the electronic weirdness that comes next on the album.
There have to be bands to differentiate between cut one, cut two and so on.
Thus, a lot of the song titles are arbitrarily stuck on, and if you were to ask Phil, "What does this title mean?" he might even have trouble identifying the song.
In a frenzied last-minute brainstorming session, we divy up the number of minutes into songs, artificially cutting it up into tracks.
In one mad swoop we come up with "Quadlibet For Tender Feet" (that would be Phil) and "The Faster We Go The Rounder We Get" (Jerry)"

The original album didn't give writing credits to individual tracks, just saying "All songs written by the Grateful
Dead." The first appearance of individual credits was on the live album
The Grateful Dead (Skull & Roses), which included a track titled
The Other One. This fades in at the end of "Cryptical Envelopment" and comprises a drum
solo followed by the "Spanish lady comes to me ..." song. It is credited to Weir and Kreutzmann, as are all
subsequent recordings of "The Other One." I used to wonder if the crediting to Kreutzmann was right:
there are a lot of interviews with Bob Weir which seem to indicate he wrote the words and music to "The Other One",
and I thought that that the credit to Kreutzmann might have arisen because the first recording on "Skull & Roses"
was preceded by a lengthy drum solo. But in Bill Kreutzmann's book "Deal" (p71) he describes how "Weir and I ...
came up with an idea that would eventually form the basis of 'The Other One'"

Confusion over the titles was sown when the first songbook "Grateful Dead Anthology" came out. The discography at the back gave writing credits as above (and for "New Potato Caboose", "Born Cross-Eyed", "Alligator" - Hunter and Petersen didn't get a mention on the original album).
But the songbook got the titles muddled and published "The other day they waited ..." as "The Other One" and "Spanish lady come to me ..." as "Cryptical Envelopment." This was the wrong way round.

The songbook "Grateful Dead Anthology II" added another element. In the discograpy, it footnoted "The Faster We Go The Rounder We Get" with "Later known as 'The Other One.'"
That is confirmed in the liner notes for "Dick's Picks 22", where the Cryptical>Other One>Cryptical suite lists the middle section as "The Faster We Go The Rounder We Get", with credits to Weir and Kreutzmann.

It seems therefore, that the original titles were arbitrary and bore only limited relation to individual songs. But the convention that has now emerged seems to be:

"Cryptical Envelopment" is the Garcia song at the beginning of the suite

"Quadlibet For Tender Feet" is the short jam between "Cryptical Envelopment" and "The Other One"

"The Faster We Go The Rounder We Get" is the Bob Weir song now known as "The Other One"

"We Leave The Castle" is the spacey jam at the end of "The Other One", with Tom Constanten's prepared piano featuring

This doesn't allow for the reprise of "Cryptical Envelopment" after "The Other One/The Faster We Go ...", but I guess there'll never be an exact match between the music and the titles.
The Dead's recordings still get confused over titles, with "Hundred Year Hall" listing "Cryptical Envelopment" when it should be "The Other One."

DeadBase says that "Willy Legate told us that in fact the title of Weir's part [ie "The Other One"] is Quodlibet For Tender Feet."
It's always possible that was once the intention, but the song-writing credits and the footnote in "Anthology II" indicate a change of mind.